To demonstrate the relationships between the gas constant R and the heat capacities C_v and C_p, we need to understand the definitions of these heat capacities and the concept of γ (gamma). In thermodynamics, C_v is the heat capacity at constant volume, while C_p is the heat capacity at constant pressure. The ratio of specific heats is defined as γ = C_p / C_v.
First, we can start from the definitions:
- C_v: The heat capacity at constant volume, represents the amount of heat required to raise the temperature of a mole of gas by one degree Celsius at constant volume.
- C_p: The heat capacity at constant pressure, represents the amount of heat required to raise the temperature of a mole of gas by one degree Celsius at constant pressure.
For an ideal gas, we can express the relationship between C_p and C_v using the following equation:
C_p – C_v = R
This equation states that the difference between the heat capacities at constant pressure and constant volume is equal to the gas constant R. Now, since we have γ defined as
γ = C_p / C_v,
we can rearrange this to find C_p:
C_p = γ * C_v.
Now we can substitute this expression for C_p into the earlier equation:
γ * C_v – C_v = R
Factoring out C_v gives us:
(γ – 1) * C_v = R
Thus, we have shown that:
R = C_v(γ – 1).
Next, we need to prove the second part of our relationships:
From the relation C_p = γ * C_v, we can express C_v as:
C_v = C_p / γ.
Using this in the earlier equation, we get:
R = C_p – C_v = C_p – (C_p / γ) = C_p(1 – 1/γ).
Rearranging this gives us:
R = C_p(γ – 1) / γ.
This confirms that:
R = C_p(γ – 1/γ).
In summary, we have shown that for an ideal gas, the relationships R = C_v(γ – 1) and R = C_p(γ – 1/γ) hold true, providing a clear understanding of how these fundamental properties of gases are interconnected.